Irrespective of the statistical packages you are using for your analyses, all you need to do first is to subject each series to unit root analyses using ADF and PP. Firstly subject raw individual series to unit root analyses using ADF and PP tests. If they are stationary without differencing, then they are I(0)s. If they are not stationary at levels, then difference each series once and conduct the unit root tests on the differenced data (series). If they are stationary after one differencing, then they are I(1)s. However, if the variable(s) are not stationary after first difference, then drop such variable(s).

Best of luck!

Dear, Saheed Busayo Akanni

Thank you for your instructive replies.

I have log-transformed the raw data (variables). Afterward, I have performed a stationarity test using ADF and PP. The variables are stationary at the first difference, meaning I(1).

My concern is-- do I have to perform a stationarity test at the second difference to make sure that those variables are NOT I(2)?

If you suppose that your series (y) may be realisations of I(2) process, test their first differences for the unit root (eg. in ADF test H0: delta y is I(1) vs H1: delta y is I(0). If you will not be able to reject the H0, treet your series (y) as I(2), otherwise test H0: y is I(1) vs H1: y is I(0)). If you star testing from the levels the results may be missleading.

Check https://www.investopedia.com/articles/trading/07/stationary.asp

Hello Sarker,

You can look at this paper.

http://fmwww.bc.edu/cfb/stata/TStalkJan2009.beamer.pdf

John C Frain

Thank you, professor. I much appreciate your instructive answer and always look for your answers in any threads of my interest.

The underlying economic theory for the series suggests that the series be I(1) and I(0). However, I will check the outcome with DF GLS variation. About cointegration, based on the result I already have after transforming (first differencing) all the series into stationary at I(1), I found no cointegration(Johansen test for cointegration), while the economic theory suggests that there shall be a long-run relationship.

Therefore I have intended to employ a VAR model to check the causal relationship and IRFs with variance decomposition.

Thank you once again.

It is difficult to show an I(2) macro-variable. what variable are looking at? Most of the time, researchers feed the data into the system and then press the button for results and then they are surprised by the results. If a variable is I(2), then you need to look at its speed of growth as an I(0). But what are you analyzing then and what interpretation do you give this I(0) variable. Danger of over-differencing.

I suggest you run a AR(k) with that variable look at the residuals. Your I(2) variable may be buffeted by shocks, if you model the shocks and justify them, you can get results that are perhaps palatable.

Provash Kumer Sarker wrote. "My concern is-- do I have to perform a stationarity test at the second difference to make sure that those variables are NOT I(2)?"

Yes, this is exactly what you need to do. You need to test the first-difference variable that you crated to prove that is stationary, and this will indicate that the original variable is I(1). If the first-difference is NOT stationary, you take a difference of the first difference and you test this second-differenced variable to see if it is stationary. If it is, then your original variable is I(2), otherwise you repeat the process until you don't find a k-difference of the original variable which is stationary. Then your var is I(k) :-)

Note: in Stata the null is NON-stationarity! So, you need values of the statistics greater than the threshold (i.e. > 1.96...) to identify a stationary variable.

Benedetto Molinari

Thank you so much. Yes, you have exactly figured it out. I have got the answer.

The variables I am working on are I(1), stationary at first difference.

Few points are mentioned.

#1. I log-transformed the raw data and then calculated the growth as I want to use the growth of the variables as my original data.

#2. All the growth variables are I(1). There is no cointegration. Therefore, I opted for a VAR model.

#3. For IRF and variance decomposition, I intend to fit the variables in growth form, i.e., the original data.

Is my concept OK? Correct me if anything goes wrong, please.

If all your growth rates (first differences of the logs of your variables) are I(1) and there is no cointegration then a VAR in the first difference of the growth rates is a valid way to proceed

it is unusual, but not impossible to find growth rates that are I(1). Some growth rates are slow to return to equilibrium values. The half-life of disequilibrium may extend over many years. Such a series may appear I(1) in short samples. Also breaks in series (e.g. changes in policy, innovations, changes in statistical definitions, changes in the base of an index calculation, etc.) may also make the series appear to be I(1). You may need to consider that such problems do not impact your analysis.

It may be difficult to interpret your model, Your IRF depends on the order that you present your series in your VAR. Different orderings may give very different interpretations. Unless you are estimating some form of SVAR you should use economic arguments to consider which order is appropriate before you estimate your VAR. Your results are then conditional on your economic arguments.

John C Frain

Dear professor, thank you once again for the intended clarification.

Yes, all my growth rates (first differences of the logs of the variables) are I(1) and there is no cointegration. Therefore, I decided to employ a VAR in the first difference of the growth rates as you mentioned.

For no cointegration in the variables, I think it is due to the sample size, (38 obs/yearly).

I used a VAR model in the first difference of the growth variables. However, for IRF analysis, I performed the VAR in the growth rates. I will cross-check the results with the first differenced variables.

It is possible that your sample size too small.

It is possible to use a VAR in I(1) variables for IRF but there must be cointegration. The theory is difficult and you must proceed very carefully if this is how you wish to proceed. Killian and Lukepohl (2017), Structural Vector Autoregressive Analysis, CUP, is a recent book-length survey that includes this material. Much of this may not be easily accessible in standard software,

If there is no cointegration the VAR in I(1) variables is probably spurious. As far as I know, there is no way to do a valid analysis with such a system.

Dear Provash.

First of all, make sure your data are from a reliable source.

Secondly, always avoid I(2) variables for analysis.

Then you can perform stationary test at 2nd difference to be sure the variables are not I(2).

Regards

You use ARDLmodel when:

- DV I(1) and IV I(0 or 1 or mixed)

We do not care DV I(2) and IV I(2).

Cong Minh Huynh

You can use ARDL when all variables are stationary. This was common in days before cointegrating.

You are confusing ARDL with the Pesaran and Shin bounds test which uses ARDL techniques to estimate various coefficients and apply the ARDL bounds to test for cointegration. The bounds test requires that

The only textbook treatment of ARDL, as far as I know, is Pesaran (2015), Time Series and Panel Data Econometrics, OUP. One should, at least, read the relevant sections of this book before using ARDL methods.

Dear Dr John C Frain

The ARDL proposed by Pesaran, Shin and Smith (2001) is in favor with 3 advantages. First, ARDL does not need all variables must be integrated at the same order, the variables can be stationary at level I(0), first difference I(1) or mix of both (Pesaran, Shin and Smith, 2001). Second, it can be used with time-series small sample data size (Hamuda, Sulikova, Gazda, and Horvath, 2013). Third, it can test the mutual relationship among variables for both short-run and long-run periods with unbiased estimation because ECM can be derived from ARDL model through a simple linear transformation (Harris and Sollis, 2003).

Best regards,

Dear Cong Minh Huynh

The three conditions that I outlined at !, 2 and 3 of my previous reply are taken from Pesaran, Shin, and Smith (2001)

The title of Pesaran, Shin, and Smith (2001) is Bounds testing approaches to the analysis of level implications. Thus I would prefer the term Bounds Test or even ARDL Bounds test.

My point is that many people who use this bounds test and quote the paper have not read the paper. For example, the paper that you quote (Hamuda et al (2013)) is an estimate of an investment model in which logINV (real gross capital formation at constant prices) is the dependent variable and log(real GDP) (gross domestic product) is one of the explanatory variables. It is extremely difficult to justify an assumption that real GDP is exogenous and the authors make no attempt to do so. To this extent, at least, the application of the bounds test is invalid. (See Pesaran (2015), pages 122 and 526. If I was a journal editor or referee I would also have saught some clarification of some other econometric points in the article

Granted that if the Psaran conditions are satisfied estimation will be efficient. If the conditions are not satisfied the results of the analysis will not be valid.

ARDL methods have been in use many many years. Have you seen the fundamental work of people like Denis Sargan, David Hendry, etc., etc, dating back about 50 years?

It is common on this and similar fora to see recommendations to use the bounds test with any regard to the possible validity of the use of the test. If you have a data set and software with an "ardl" instruction it is not valid to use the instruction. One must consider both the underlying economic theory and the requirements for the validity of the test.

Why would you suspect that your variables are integrated at order 2, are the results of unit root tests at order 1 of integration inconclusive? if that is the case, then there is a good reason to believe that your variables are integrated at order 2. You should run many stationarity tests to make sure that your results are robust. Macroeconomic variables are usually integrated at order 1, it rarely shows an integration of order 2.

Also, I am not surprised that you could not find any evidence of cointegration among the variables, and that is highly likely due to the small size of your data sample. I had the same problem in one of my studies, my data sample was as small as yours. Small data samples are usually not reliable due to issues with the degree of freedom and t statistics. In my case, I could not enlarge the data sample, so I had only one solution which was to remove some explanatory variables, the least to be impactful theoretically, and that worked for me. I don't know how many variables you are using in your study, but you should consider removing some variables if you cannot add more observations to your small data sample.

Hope this help